Relating field theories via stochastic quantization

This note aims to subsume several apparently unrelated models under a common framework. Several examples of well-known quantum field theories are listed which are connected via stochastic quantization. We highlight the fact that the quantization method used to obtain the quantum crystal is a discrete analog of stochastic quantization. This model is of interest for string theory, since the (classical) melting crystal corner is related to the topological A-model. We outline several ideas for interpreting the quantum crystal on the string theory side of the correspondence, exploring interpretations in the Wheeler–De Witt framework and in terms of a non-Lorentz invariant limit of topological M-theory.     

Topological quantization and cohomology

The relationships between topological charge quantization, Lagrangians and various cohomology theories are studied. A very general criterion for charge quantization is developed and applied to various physical models. The relationship between cohomology and homotopy is discussed.This work was supported in part by the National Science Foundation under Contracts PHY 81-18547; and by the Director, Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Contracts DE-AC03-76SF00098Alfred P. Sloan Foundation Fellow     

Topological quantization in units of the fine structure constant

Fundamental topological phenomena in condensed matter physics are associated with a quantized electromagnetic response in units of fundamental constants. Recently, it has been predicted theoretically that the time-reversal invariant topological insulator in three dimensions exhibits a topological magnetoelectric effect quantized in units of the fine structure constant α=e2/?c. In this Letter, we propose an optical experiment to directly measure this topological quantization phenomenon, independent of material details. Our proposal also provides a way to measure the half-quantized Hall conductances on the two surfaces of the topological insulator independently of each other.     

Higher algebraic structures and quantization

We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in ad+1 dimensional topological theory to manifolds of dimension less thand+1. We then construct a generalized path integral which ind+1 dimensions reduces to the standard one and ind dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.The author is supported by NSF grant DMS-8805684, a Presidential Young Investigators award DMS-9057144, and by the O'Donnell Foundation. He warmly thanks the Geometry Center at the University of Minnesota for their hospitality while this work was undertaken     

Hamiltonian BRST quantization of the conformal string

We present a new formulation of the tensionless string (T = 0) where the space-time conformal symmetry is manifest. Using a Hamiltonian BRST scheme we quantize this Conformal String and find that it has critical dimension D = 2. This is in keeping with our classical result that the model describes massless particles in this dimension. It is also consistent with our previous results which indicate that quantized conformally symmetric tensionless strings describe a topological phase away from D = 2.

We reach our result by demanding nilpotency of the BRST charge and consistency with the Jacobi identities. The derivation is presented in two different ways: in operator language and using mode expansions.

Careful attention is paid to regularization, a crucial ingredient in our calculations.     

Topological quantization of disclination points in three—dimensional liquid crystals

Using the φ-mapping method and topological current theory, we study the inner structure of disclination points in three-dimensional liquid crystals. By introducing the strength density and the topological current of many disclination points, it is pointed out that the disclination points are determined by the singularities of the general director field and they are topologically quantized by the Hopf indices and Brouwer degrees.     

Geometric quantization of topological gauge theories

We show that Abelian gauge theories in 2+1 space-time dimensions with the introduction of a topological Chern-Simons term can be quantized with the use of the symplectic formalism. The consistency of our results are verified by the agreement with the ones from the Dirac case.     

Soliton quantization in lattice field theories

Quantization of solitons in terms of Euclidean region functional integrals is developed, and Osterwalder-Schrader reconstruction is extended to theories with topological solitons. The quantization method is applied to several lattice field theories with solitons, and the particle structure in the soliton sectors of such theories is analyzed. A construction of magnetic monopoles in the four-dimensional, compactU(1)-model, in the QED phase, is indicated as well.     

Semiclassical quantization with a quantum adiabatic phase

By applying the path integral method to two interacting systems, we derive a novel form of the semiclassical quantization rule including the topological phase which was recently found in certain quantum adiabatic processes.     

Landau quantization for an induced electric dipole in the presence of topological defects

In this contribution we investigate the non-relativistic quantum dynamics of induced electric dipoles in the presence of a topological defect. We propose an analog of Landau quantization for neutral atoms, where a electric dipole is induced by the electromagnetic field configuration. We investigate this system in the presence of a topological defect and show that it breaks the infinite degeneracy of Landau levels.     

Landau quantization in the spinning cosmic string spacetime

We analyze the quantum phenomenon arising from the interaction of a spinless charged particle with a rotating cosmic string, under the action of a static and uniform magnetic field parallel to the string. We calculate the energy levels of the particle in the non-relativistic approach, showing how these energies depend on the parameters involved in the problem. In order to do this, we solve the time independent Schrödinger equation in the geometry of the spinning cosmic string, taking into account that the coupling between the rotation of the spacetime and the angular momentum of the particle is very weak, such that makes sense to apply the Schrödinger equation in a curved background whose metric has an off diagonal term which involves time and space. It is also assumed that the particle orbits sufficiently far from the boundary of the region of closed timelike curves which exist around this topological defect. Finally, we find the Landau levels of the particle in the presence of a spinning cosmic string endowed with internal structure, i.e., having a finite width and uniformly filled with both material and vacuum energies.     

Universal metaplectic structures and geometric quantization

The recently developed concepts of generalized and universal spin structures are carried over from the orthogonal to the symplectic and unitary cases. It turns out that the analogues ofSpin ^c-structures, namely theMp ^c-structures andMU ^c-structures, are sufficient to avoid topological obstructions to their existence. It is indicated how this fact can be used in the geometric quantization of certain suitably polarized symplectic manifolds with arbitrary second Stiefel-Whitney class, where the usual Kostant-Souriau quantization scheme breaks down.Supported by the Studienstiftung des deutschen Volkes     

Topological invariant and the quantization of the Hall conductance

The topological aspects of wavefunctions for electrons in a two dimensional periodic potential with a magnetic field are discussed. Special attention is paid to the linear response formula for the Hall conductance σ_xy. It is shown that the quantized value of σ_xy is related to the number of zeros of wavefunctions in the magnetic Brillouin zone. A phase of wavefunctions cannot be determined in a unique and smooth way over the entire magnetic Brillouin zone unless the magnetic subband carries no Hall current.     

Canonical quantization of general relativity in discrete space-times

It has long been recognized that lattice gauge theory formulations, when applied to general relativity, conflict with the invariance of the theory under diffeomorphisms. We analyze discrete lattice general relativity and develop a canonical formalism that allows one to treat constrained theories in Lorentzian signature space-times. The presence of the lattice introduces a "dynamical gauge" fixing that makes the quantization of the theories conceptually clear, albeit computationally involved. The problem of a consistent algebra of constraints is automatically solved in our approach. The approach works successfully in other field theories as well, including topological theories. A simple cosmological application exhibits quantum elimination of the singularity at the big bang.     

Fuzzy topology, quantization and gauge invariance

Quantum space-time with Dodson-Zeeman topological structure is studied. In its framework the states of massive particle m correspond to elements of fuzzy set called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty σ _x . Quantization formalism is derived from consideration of m evolution in fuzzy phase space with minimal number of additional assumptions. Particle’s interactions on fuzzy manifold are studied and shown to be gauge invariant.     

Landau quantization and the thickness limit of topological insulator thin films of Sb2Te3

We report the experimental observation of Landau quantization of molecular beam epitaxy grown Sb{2}Te{3} thin films by a low-temperature scanning tunneling microscope. Different from all the reported systems, the Landau quantization in a Sb{2}Te{3} topological insulator is not sensitive to the intrinsic substitutional defects in the films. As a result, a nearly perfect linear energy dispersion of surface states as a 2D massless Dirac fermion system is achieved. We demonstrate that four quintuple layers are the thickness limit for a Sb{2}Te{3} thin film being a 3D topological insulator. The mechanism of the Landau-level broadening is discussed in terms of enhanced quasiparticle lifetime.     

Topological quantization of current in quantum tunnel contacts

It is shown that an account of the Berry phase (a topological ϑ-term), together with a dissipative term in the effective action S[ϕ] of the tunnel contacts, induces a strong quantization of the tunnel current at low temperatures. This phenomenon, as the Coulomb blockade, reflects a discrete charge structure of the quantum shot noise and can ensure a quantization of the tunnel current without a capacitive charging energy E _C when the latter is strongly suppressed by quantum fluctuations. Since the value of the ϑ parameter is determined by the gate voltage, this effect allows us to control the current through the contact. A possible physical application of this effect is proposed. The text was submitted by the author in English.     

Flux quantization and the M-theoretic characters

In a previous work [H. Sati, M-theory and characteristic classes, JHEP 0508 (2005) 020, hep-th/0501245] we introduced characters and classes built out of the M-theory four-form and the Pontrjagin classes, which we used to express the Chern–Simons and the one-loop terms in a way that makes the topological structures behind them more transparent. In this paper we further investigate such classes and the corresponding candidate generalized cohomology theories. In particular, we study the flux quantization conditions that arise in this context.     

Topological Quantization of the Magnetic Flux

The quantization of the magnetic flux in superconducting rings is studied in the frame of a topological model of electromagnetism that gives a topological formulation of electric charge quantization. It turns out that the model also embodies a topological mechanism for the quantization of the magnetic flux with the same relation between the fundamental units of magnetic charge and flux as there is between the Dirac monopole and the fluxoid.     

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.